Stew:
... this error "expected initializer before '+=' token"
is pretty clear... look here:
String serialInput += (char)Serial.read();
your compiler expected an initializer...
String serialInput = ""; // initialize a String called serialInput
serialInput += (char)Serial.read(); // append the string
but you still have a lot of issues:
Serial.print( int serialInput); // what the heck is this, for example?